An Optimal Decomposition Algorithm for Tree Edit Distance

The edit distance between two ordered rooted trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this article, we present a worst-case O(n(3))-time algorithm for the problem when the two trees have size n, improving the previous best O(n(3) log n)-time algorithm. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms-which also includes the previous fastest algorithms-by tightening the known lower bound of Omega(n(2) log(2) n) to Omega(n(3)), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds for decomposition strategy algorithms of Theta(nm(2)(1+ log n/m) when the two trees have sizes m and n and m < n.